A spaceship orbits around a planet at a height of \(20\,km\) from its surface. Assuming that only gravitational field of the plant acts on the spaceship. What will be the number of complete revolutions made by the spaceship in \(24\,hours\) around the plane? [Given: Mass of plane \(= 8 \times 10^{22}\,kg,\) Radius of planet \(= 2\times 10^6\,m,\) Gravitational constant \(G = 6.67\times 10^{-11}\,Mn^2/kg^2\) ]

\(V = \sqrt {\frac{{GM}}{r}} \)

\(n = \frac{{VT}}{{2\pi r}} = \sqrt {\frac{{GM}}{r}} \frac{T}{{2\pi r}}\)

\( = \left( {\sqrt {\frac{{GM}}{{{r^3}}}} } \right) \times \frac{T}{{2\pi }} = \sqrt {\frac{{6.67 \times {{10}^{ - 11}} \times 8 \times {{10}^{22}}}}{{{{\left( {202 \times {{10}^4}} \right)}^3}}}}  \times \frac{T}{{2\pi }}\)

\( = \frac{{24 \times 3600}}{{2 \times 3.14}}\sqrt {\frac{{6.67 \times 8 \times {{10}^{11}}}}{{{{\left( {202} \right)}^3} \times {{10}^{12}}}}} \)

\( = \frac{{24 \times 3600}}{{2 \times 3.14 \times 1242.8}} = \frac{{24 \times 3600}}{{78.51}} \simeq 11\)